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A307532 a(n) is the smallest k > 2^(2^n)+1 such that 2^(k-1) == 1 (mod (2^(2^n)-1)*k). 1
5, 7, 29, 281, 65617, 4294967681, 18446744073709552577, 340282366920938463463374607431768211841, 115792089237316195423570985008687907853269984665640564039457584007913129642241 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

a(n) is smallest k > 2^(2^n)+1 such that k == 1 (mod 2^n) and 2^(k-1) == 1 (mod k), so a(n) is an odd prime or a Fermat pseudoprime to base 2.

a(n) is the least k = 2^(2^n) + m*2^n + 1 for m > 0 such that 2^(k-1) == 1 (mod k).

The values of m = (a(n)-2^(2^n)-1)/2^n are 2, 1, 3, 3, 5, 12, 15, 3, 9, 202, 56, 304, 635, 11095, 8948, ...; is m = A307535(n) for all n > 4?

Conjecture: a(n) is prime for all n >= 0.

LINKS

Table of n, a(n) for n=0..8.

FORMULA

a(n) == 1 (mod 2^n).

MATHEMATICA

a[n_] := Module[{k = 2^(2^n) + 2}, While[PowerMod[2, k - 1, (2^(2^n) - 1)*k] != 1, k++]; k]; Array[a, 10, 0]

PROG

(PARI) a(n) = my(k=2^(2^n)+2);  while( Mod(2, (2^(2^n)-1)*k)^(k-1) != 1, k++); k; \\ Michel Marcus, Apr 25 2019

CROSSREFS

Cf. A001567, A065091, A307512, A307535.

Sequence in context: A171619 A153411 A081630 * A135324 A107639 A069688

Adjacent sequences:  A307529 A307530 A307531 * A307533 A307534 A307535

KEYWORD

nonn

AUTHOR

Amiram Eldar and Thomas Ordowski, Apr 13 2019

EXTENSIONS

a(8) from Chai Wah Wu, Apr 29 2019

STATUS

approved

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Last modified August 9 02:14 EDT 2020. Contains 336310 sequences. (Running on oeis4.)